The Dimensional Recurrence and Analyticity Method for Multicomponent Master Integrals: Using Unitarity Cuts to Construct Homogeneous Solutions

We consider the application of the DRA method to the case of several master integrals in a given sector. We establish a connection between the homogeneous part of dimensional recurrence and maximal unitarity cuts of the corresponding integrals: a maximally cut master integral appears to be a solution of the homogeneous part of the dimensional recurrence relation. This observation allows us to make a necessary step of the DRA method, the construction of the general solution of the homogeneous equation, which, in this case, is a coupled system of difference equations.Comment: 17 pages, 2 figure

Super congruences and Euler numbers

Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\binom{2k}{k}/2^k=(-1)^{(p-1)/2}-p^2E_{p-3} (mod p^3),$$ $$\sum_{k=1}^{(p-1)/2}\binom{2k}{k}/k=(-1)^{(p+1)/2}8/3*pE_{p-3} (mod p^2),$$ $$\sum_{k=0}^{(p-1)/2}\binom{2k}{k}^2/16^k=(-1)^{(p-1)/2}+p^2E_{p-3} (mod p^3)$$, where E_0,E_1,E_2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for $\pi^2$, $\pi^{-2}$ and the constant $K:=\sum_{k>0}(k/3)/k^2$ (with (-) the Jacobi symbol), two of which are $$\sum_{k=1}^\infty(10k-3)8^k/(k^3\binom{2k}{k}^2\binom{3k}{k})=\pi^2/2$$ and $$\sum_{k>0}(15k-4)(-27)^{k-1}/(k^3\binom{2k}{k}^2\binom{3k}k)=K.$

Escherichia coli Bacteriocins: Antimicrobial Efficacy and Prevalence among Isolates from Patients with Bacteraemia

Bacteriocins are antimicrobial peptides generally active against bacteria closely related to the producer. Escherichia coli produces two types of bacteriocins, colicins and microcins. The in vitro efficacy of isolated colicins E1, E6, E7, K and M, was assessed against Escherichia coli strains from patients with bacteraemia of urinary tract origin. Colicin E7 was most effective, as only 13% of the tested strains were resistant. On the other hand, 32%, 33%, 43% and 53% of the tested strains exhibited resistance to colicins E6, K, M and E1. Moreover, the inhibitory activity of individual colicins E1, E6, E7, K and M and combinations of colicins K, M, E7 and E1, E6, E7, K, M were followed in liquid broth for 24 hours. Resistance against individual colicins developed after 9 hours of treatment. On the contrary, resistance development against the combined action of 5 colicins was not observed. One hundred and five E. coli strains from patients with bacteraemia were screened by PCR for the presence of 5 colicins and 7 microcins. Sixty-six percent of the strains encoded at least one bacteriocin, 43% one or more colicins, and 54% one or more microcins. Microcins were found to co-occur with toxins, siderophores, adhesins and with the Toll/Interleukin-1 receptor domain-containing protein involved in suppression of innate immunity, and were significantly more prevalent among strains from non-immunocompromised patients. In addition, microcins were highly prevalent among non-multidrug-resistant strains compared to multidrug-resistant strains. Our results indicate that microcins contribute to virulence of E. coli instigating bacteraemia of urinary tract origin